The mathematics of Safety Abstract
This module addresses the topics of measures of central tendency and spread. The instructional objectives are to summarize and analyze univariate data using statistical analysis and graphs. (NCSCOS Objective#1.02: Summarize and analyze univariate data to solve problems. Advanced Functions and Modeling; #2.01: Describe data to solve problems, Discrete Math)
The context of the module is examining safety concerns due to noise exposure on construction sites, and possible hearing damage.
This module is intended for Advanced Functions and Modeling, which has Algebra 2 as a prerequisite. However, it could be adapted for use in any course in which statistical analysis of data is covered- Discrete Math, for example.
This module will build on student knowledge of data analysis. It will span two to three 90-minute class periods, which includes time in the computer lab for students to graph their results. Some of the work will be done by students outside the classroom as students collect the necessary data.
The module is project based and utilizes technologies including the graphing calculator, TI Interactive or Microsoft Excel, and a Dosimeter (a device that records noise levels in decibels). (Note: Vernier also makes a sound meter which records in decibels to the graphing calculator. See http:www.vernier.com/probes/slm-bta.html for more information.) At the end of the module the students will submit a report containing the results of their analysis and computer generated graphs.
The unit will be developed with the assistance of Mr. Jon Yuhas of The Roberts Construction Company. The Roberts Company constructs and maintains large pressurized tanks. The company is located in Winterville, NC and does business worldwide. Mr. Yuhas is a Vice President of the company and is responsible for, among other things, maintaining safety standards. Students will analyze data that have been collected recording the noise levels that workers are exposed to on the job. The data were collected by a device called a dosimeter that was attached to a worker and measured the noise level in decibels that the worker was exposed to during the course of a day. The average decibel level was recorded at five-minute intervals. Students will investigate the data, using numerical analysis and graphs, and compare it to the acceptable standards proscribed by OSHA. Students will also have the opportunity to use the dosimeter to collect data during the school day to determine the noise levels to which they’re exposed, after first having predicted where in the school building their hearing might be most at risk.
The topic of noise levels and decibels will be revisited later in the course when students study logarithms, as decibels are a logarithmic scale. At that time we will solve real world problems and compare different levels of noise exposure. (For example, how the noise level in the gym during a basketball game compares to that of a jet engine.)
The Mathematics of Safety Outline
Teacher Preparation:
Materials- Student Project and Notes
Sets of data collected on site at Roberts Company
Technology-Graphing Calculator
TI Interactive or Microsoft Excel
Dosimeter (optional)
Key terms- Decibels
Occupational Safety and Health Administration (OSHA)
Central tendency
Measures of Spread
Displays of data
Essential Question:
To what extent is your hearing at risk if you work for a construction company such as Roberts Construction in Winterville?
Supporting Questions:
What limit is set by OSHA concerning noise levels in the workplace?
What measures do Roberts Company take to be sure that they are in compliance with OSHA standards?
What noise are students exposed to on a daily basis that could endanger their hearing?
How is noise level measured?
Project Timeline:
1. Days prior to the project-review measures of central tendency and spread, ways to display data, and how to use the graphing calculator to perform statistical analysis. (Two to three 90-minute class periods)
2. Student introduction to The Mathematics of Safety.
3. Student analysis of data collected at Roberts Construction site. (Done in groups using the graphing calculator, with each group analyzing a different set of data) (One 90-minute period for steps 2&3 combined.)
4. Students introduced to TI Interactive or Microsoft Excel to graphically display data. Students given time in the computer lab to generate their report.(One 90-minute class period)
5. (Optional/Independent student work) Students predict where at school their hearing might be at risk. Then they collect their own data and analyze it regarding their hearing health.
Alternative to step 5: Students research the decibel levels of things to which they are exposed and see if these things fall within OSHA standards for maintaining hearing health. ( Suggested site- http://www.sfu.ca/sonic-studio/handbook/Decibel.html)
The Mathematics of Safety Project
According to OSHA guidelines, “the employer shall administer a continuing, effective hearing conservation program…whenever employee noise exposures equal or exceed an 8-hour time-weighted average sound level of 85 decibels”. (www.osha.gov) Each group of students will be given data collected by an employee of the Roberts Company that reports the decibel levels of noise to which they were exposed during a work day. The data reports the average decibel level for each five minute interval during the day. Your task is to analyze the data as described below, and determine whether this employee requires a hearing conservation program according to the OSHA guideline described above. The numerical and graphical analysis may be done as a group, but each group member is responsible for writing his own summary and completing the ‘Above and Beyond’ section.
Project Requirements (Point values are in parentheses):
(10) Numerical Analysis of data- including measures of central tendency and spread, the five number summary, the percentile rank for 85, the middle 50%of the data, and the percent of the data that is within one standard deviation of the mean. This may be done on the graphing calculator and typed, or may be computer generated. Each statistic must be clearly labeled.
(5) Graphical Display of the data- this must be computer generated and clearly labeled, or no points will be awarded for this component. Your project should contain a histogram of the decibel levels, and a scatter plot of decibel level vs. time. The graphs should show an accurate and appropriate scale.
(5) Summary-remember, this part must be done individually, or no points will be awarded for this component. Your summary should appropriately refer to the OSHA standard described above, your numerical analysis and both your graphs. Additionally, you must clearly and correctly conclude whether your employee requires a hearing conservation plan.
(2) Above and Beyond- remember, this is also to be done individually. To earn credit for this component, you must obtain a copy of the data collected at the school and, using numerical and graphical analysis, determine whether the student involved requires a hearing conservation plan. Be sure to refer to your statistics and graph and clearly state your conclusion.
The Mathematics of Safety Assessment Sheet
Numerical analysis (10):
Measures of central tendency (3) ______
Measures of Spread (3) ______
One St. Dev. from the mean ______
Middle 50% of data ______
5 # Summary ______
Percentile rank, 85 ______
Graphical display (5):
Computer generated ______
Clearly labeled ______
Histogram ______
Scatter plot ______
Summary (5):
Appropriate reference to numerical analysis ______
Appropriate reference to graphs ______
Correct & clearly expressed conclusion ______
Above and Beyond (2):
Numerical analysis and graph ______
Conclusion ______
The Mathematics of Safety Teacher Notes
Purpose: The purpose of the project is to enable students to analyze real world data numerically and graphically and draw conclusions based on their statistics and graphs. (NC SCOS for AFM, goal 1.02) Additionally students are expected to use appropriate technology to produce computer-generated graphs.
Company information: The Roberts Company custom fabricates and offers services in a variety of areas, including large pressurized tanks and modular construction. They are located in Winterville, NC and manufacture items that are shipped worldwide. My business partner is Mr. Jon Yuhas, and one of his areas of expertise is employee safety. Employees at the Roberts Company are routinely asked to wear a device called a Dosimeter that measures the decibel levels of the noise to which they are exposed throughout the day. The device continuously records noise levels, and the associated print-out summarizes the averages at five minute intervals. Additionally, workers on the shop floor are required to wear ear plugs that provide additional protection (33dB) against hearing loss. To obtain more information about the Roberts Company, you may visit their web site at www.robertscompany.com.
Technology used: Students will use a graphing calculator to perform necessary statistical calculations. They will use TI Interactive, Microsoft Excel, or some other appropriate software to generate their graphs. Refer to student notes and Appendix A for step by step instructions on generating graphs using Microsoft Excel. As an optional component, students will use a dosimeter or similar equipment to collect data on environmental noise levels.
Occupational Safety and Health Administration (OSHA) Standards: (www.OSHA.gov) The government web site information is difficult to interpret, but according to my business partner basically requires that employees must be provided with a hearing conservation plan if they are exposed to an average of 85 decibels (dB) of noise or more per day. The ear protectors that Roberts Company uses provide protection for up to 33 dB above this, suitable for an average daily exposure of less than 118 dB. Data are collected frequently at the company to document compliance with OSHA standards. Note: OSHA is only one of several organizations to which Roberts Company is accountable.
Decibel Scale of Measurement: Sound power level in decibels (dB) is a logarithmic unit used to describe the ratio of sound intensity (in watts/m2 ) to the lowest sound people of excellent hearing can hear, N0.
L=Sound power level (dB)
N=Sound power (watts/m2)
N0=10-12 watts/m2
L=10log(N/ N0)
ex: A jet aircraft has sound power of 105watts/m2
L=10log(105/10-12)
=10log(1017)
=170 dB
Suggested timeline:
1. Review the concepts: measures of central tendency, measures of spread, graphical displays of data, and use of a graphing calculator to facilitate these. (90-180 min)
2. Introduction of project including project requirements, discussion of decibel scale and hearing conservation needs while working construction, and group assignments. (20-30 min)
3. Student group work on project including numerical analysis on the graphing calculator.(45-60 min)
4. Group and Individual work on project in the computer lab, including necessary instruction on software use. (60-90 min)
5. Independent student work as necessary to complete the project.
The Mathematics of Safety Student Notes
These notes contain step by step instructions for those of you who are unfamiliar with Microsoft Excel. They will also help you to organize the information you will need to successfully complete your project.
In the Classroom
1. With your group members, input the data on decibel level into L1 in your calculator. Record the data set number here:
______
2. Calculate the one-variable statistics and record your results here:
n= ______Q1 ______
Mean=______Med______
St. Dev=______Q3______
Min=______Max______
3. Sort the data. Then calculate determine the mode(s) and percentile rank for 85 dB.
Mode ______percentile rank ______
4. On the calculator, view the histogram and adjust the window to find “friendly”
values for the xmin, xmax and xscale.
Xmin______Xmax______Xscale______
In the Computer lab
5. Log on and open a Word document. Type in a heading that includes your name, subject, class period and data set number. Then type in the result of your numerical analysis, clearly labeling each statistic. Refer to your project sheet to make sure you include all that is required.
6. Open an Excel spreadsheet. Then open my Excel sheet and find and highlight your data set. Copy and paste it into your spreadsheet in columns A&B.
7. The steps below will lead you through the creation of a histogram graph for the 2003 version of MS Excel. For MS Excel 2007, see appendix A.
The first thing you need to do is create the values for your x-axis in column C of your Excel spreadsheet. Then you will probably need to add the data analysis package to your “Tools” menu. Then you will be able to draw the graph.
In column C, type in numbers from your Xmin value to your Xmax value in increments of your Xscl.
From the Excel toolbar, choose “Tools/data analysis/histogram” and click OK. (If “data analysis” isn’t showing under the Tools menu, you must first select “Tools/add-ins” and check “Analysis toolpak” and click OK.)
With your cursor in the “Input Range” box, highlight all of column B data from top to bottom (the decibel levels). Then move your cursor to “Bin Range” and highlight all of your column C data.
Under “Output options” check “New Worksheet Ply” and type in the name “Histogram”. Also select “Chart output”.
If you have followed these steps you should now have a histogram graph showing when you click on the “Histogram” tab at the bottom of your spreadsheet. You need to make a few changes before putting it in your Word document. You must add labels, remove the gaps between the bars, and remove the background color if you’re printing in black and white.
Right click on one of the histogram bars and select “Format data series”. Choose the “options” tab and change the “Gap width” to 0 (zero). You may also want to choose the “patterns” tab and “fill effects” and select a pattern. Click OK.
Return to the histogram graph. Right click on the histogram background area and select “Format plot area”. Choose “area” and “color”, click “none” and highlight the color white. Click OK.
Return to the histogram graph. Click on each label on your graph and type in the appropriate label. Then if necessary, resize your graph to enhance its appearance. Your histogram should now be ready to copy and paste into your Word document.
8. The following steps will lead you through graphing your scatter plot in Excel. It is considerably simpler than the histogram!
Return to your Excel document. Highlight all your data in columns A and B. (time and dB levels)
From the Excel toolbar, select “Chart Wizard”, choose the “standard type” tab and highlight “xy scatter”. Click “next” two times. Type in the graph labels and click “next”. Check “as new sheet” and type “scatter”, click “finish”.